Question

a population of e. coll bacteria grows exponentially with time. you believe that the mean time between divisions is ( t_{d}=29 ) min, and that cell death occurs on average after ( t_{m}=200 mathrm{hr} ). the population starts with 1000 cells. complete parts (a) through (c). (a) use the equation ( n(t)=n_{0} e^{(b - m) t} ) to predict how many cells are present after 4 hours. the number of cells present after 4 hours is ( square ). (do not round until the final answer. then round to the nearest integer as needed.)

Explanation:

Step1: Convert \(t_b\) and \(t_m\) to the same unit

Since \(t_b = 29\) min and \(t_m=200\) hr. Convert \(t_b\) to hours: \(t_b=\frac{29}{60}\) hr. Then \(b - m=\frac{1}{t_b}-\frac{1}{t_m}=\frac{60}{29}-\frac{1}{200}\).

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Step2: Substitute into the formula

We know \(N_0 = 1000\), \(t = 4\) hr. Using the formula \(N(t)=N_0e^{(b - m)t}\), substitute \(N_0 = 1000\), \(t = 4\) and \(b - m\approx2.064\) into it.

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Since \(e^{8.256}\approx3860.9\) (using a calculator for the exponential function \(y = e^x\) where \(x = 8.256\))

$$ N(4)=1000\times3860.9 = 3860900 $$

Answer:

\(3860900\)